There are many booklets and articles that discuss this subject in detail, such as:
My answer here is just a brief summary of the general ideas.
Genesis 2:3 says that "God blessed the seventh day, and sanctified it".
The Bible talks about this day in terms of God, not in terms of a specific covenant or nation.
Similarly, the seven annual holy days are presented as events that are sacred to God, not as events that are part of a specific covenant with a specific people (except in the way God's chosen people are representative of all mankind).
The feasts of the LORD, which you shall proclaim to be holy convocations, these are My feasts.
Six days shall work be done, but the seventh day is a Sabbath of solemn rest, a holy convocation. You shall do no work on it; it is the Sabbath of the LORD
… These are the feasts of the LORD
- … the LORD’s Passover.
- … the Feast of Unleavened Bread to the LORD
… bring a sheaf of the firstfruits of your harvest to the priest.
He shall wave the sheaf before the LORD, to be accepted on your behalf; on the day after the Sabbath the priest shall wave it.
- … Count fifty days to the day after the seventh Sabbath; then you shall offer a new grain offering to the LORD.
- … In the seventh month, on the first day of the month, you shall have a sabbath-rest, a memorial of blowing of trumpets, a holy convocation. … you shall offer an offering made by fire to the LORD
- … the Day of Atonement, to make atonement for you before the LORD your God.
- … the Feast of Tabernacles for seven days to the LORD.
- … On the eighth day you shall have a holy convocation, and you shall offer an offering made by fire to the LORD. It is a sacred assembly
… These are the feasts of the LORD which you shall proclaim to be holy convocations, to offer an offering made by fire to the LORD, a burnt offering and a grain offering, a sacrifice and drink offerings, everything on its day
— Leviticus 23
Notice that it is the Sabbath and the seven festivals themselves that are holy or sacred, that they are "the feasts of the LORD", not "of Israel" and certainly not "of the Jews".
The people of Israel might have been chosen to know about these festivals, just as they were chosen to know about God's other commandments and laws, but God's holy days were not created for the Israelites only.
Just as ("The Sabbath was made for man, and not man for the Sabbath" — Mark 2:27), God's festivals were create for man, all mankind.
Later, when "the LORD" was incarnated as Jesus, he warned people:
Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill.
For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled.
— matthew 5:17–18
The Gospels record specific instances of Jesus's celebrating some of God's feasts.
And we can in fact be certain that had he always celebrated all of them: had he ever failed to do so, the Pharisees would have made a far far greater fuss about it than they did for something as simple of ceremonial hand washing.
Following the Crucifixion, the Bible records that Christians continued to celebrate God's festivals.
- I must by all means keep this coming feast in Jerusalem (Acts 18:21).
- Therefore let us keep the feast, not with old leaven, nor with the leaven of malice and wickedness, but with the unleavened bread of sincerity and truth. (1 Corinthians 5:8).
Except for the obvious exception of Hebrews, Paul's epistles were mostly written to groups mostly composed of gentile converts, not Jews.
In them, he frequently warns these new Christians to continue celebrating God's Sabbath and holy days, ignoring the criticism of their still-pagan friends, relatives, and co-workers, who as ascetics etc. reject celebration and pleasure, and therefore appear to have a higher moral standard.
(E.g. see my answers to Does Paul reject the idea of celebrating holidays in Galatians? and Keeping the Sabbath and Colossians 2:16.)
And Zechariah 14:19 prophesies that in the future, in the Kingdom of God, all nations will celebrate God's festivals: "This shall be the punishment of Egypt and the punishment of all the nations that do not come up to keep the Feast of Tabernacles.".
It's obvious that the early Christian Church celebrated God's holy days, which were created for all mankind, and that they will continue to be celebrated during the Millennium.
The original question has been answered, but there is something far more significant to this than the continuation of harvest rituals and the commemoration of historical events.
Consider the physical meanings of God's holy days:
- At the original Passover, the blood of the sacrificial lamb saved the Israelites from death.
- During the Days of Unleavened Bread, the Israelites removed all traces of yeast from their households, in memory of when they left Egypt in the Exodus.
The Wave-sheaf ceremony, on the first day after the first weekly Sabbath during this week, celebrates the beginnings of the spring harvest.
- Pentecost, 50 days after that first-fruits ceremony, is traditionally considered to be when God presented Moses with the Ten Commandment tablets, symbolizing the official beginning of the Israelites being given God's true religion.
- The Feast of Trumpets is the beginning of the civil year, with trumpets traditionally being used to signal danger or important announcements.
It is considered a wake-up-call to remember God and to review one's behaviour.
- The Day of Atonement is the most solemn day of the year, a day of fasting and contemplation of one's relationship with God.
- During the Feast of Tabernacles one commemorates the years in the wilderness following the Exodus by living in tents or other temporary dwellings.
Families would typically travel to larger cities, in particular Jerusalem, to celebrate and enjoy the week, having set aside 10% of their yearly income for this event, much like a modern summer vacation.
- The Last Great Day immediately follows the Feast of Tabernacles.
It is a solemn day of public Bible reading and a confirmation of knowledge of God's laws.
But these holy days also serve as symbolic prophecies of God's plan for mankind.
The Spring holy days represent what has already happened:
- It was at Passover, that the lamb of God shed his blood to save people from death, to provide Salvation.
- The Days of Unleavened Bread symbolize what a Christian must do in accepting Christ's sacrifice.
Egypt is a type of sin, with the Exodus representing repentance and separation from worldly sin, and the yeast demonstrating how accepting even a small amount of sin can allow it to quickly spread throughout one's life.
The Wave-sheaf first-fruits ceremony represents Jesus's being the first of many to be resurrected to eternal life.
It was on the day after the first weekly Sabbath of the Days of Unleavened Bread that Jesus first ascended to Heaven.
(In the morning he said not to touch him because he had not yet ascended, but later that same day allowed physical contact.)
- Pentecost symbolizes the founding of the Christian Church, and the acceptance of God's holy spirit.
The Fall holy days represent what is about to happen:
- The Feast of Trumpets is a prophecy of Christ's return to Earth, to save mankind from the mess it has made, and the first general resurrection of those that have been saved.
("And He will send His angels with a great sound of a trumpet, and they will gather together His elect from the four winds, from one end of heaven to the other." — Matthew 24:31)
- The Day of Atonement symbolizes the binding of Satan for a thousand years, allowing God and mankind to live as one in the Kingdom of God on Earth.
- The Feast of Tabernacles symbolizes Jesus's temporary return to Earth, away from his Heavenly home.
It is a time of great joy and celebration, with mankind being taught God's way of life free from Satan's influence, and with full historical knowledge of how badly mankind ruined the Earth by following its own ideas.
- The Last Great Day symbolizes the judgement period at the end of the Millennium.
All those that have died without a chance of salvation will be part of the second general resurrection.
They will all be taught God's way, and almost all will accept it.
At the end, the few that continue to reject God's spirit (the unpardonable sin) will be mercifully destroyed.
During this whole process, God has been preparing mankind to develop perfect character, symbolically harvesting them to literally become his children, brothers of Jesus, like him in every way.
The booklets listed at the top (and many similar publications) provide much more detailed descriptions of how God's holy days symbolize God's plan for mankind.
I also wrote my own summary more than a decade ago: A Very Short History of the World — a Biblical Christian View.